Exploring the Space of Human Interaction

In the book Theoretical Morphology, George McGhee examines why living things look the way they do. He explores the space of the potential shapes of organisms, or their morphology, and compares that to what we find in Nature, finding that the actual morphologies are often a subset of those potential shapes, due to chance and selection.

For example, the morphological space of certain types of mollusc shells can be described by two simple parameters:

However, the actual distribution of shell shapes is denser in some areas of the morphological space, and entirely absent in others:

Recently, a team of researchers at Cornell University and Facebook set out to see if a similar kind of morphological space could be explored for the space of social interactions. About fifty years ago, a sociologist found hints of the limits on the structure of social networks in a study of children:

Testing friendship relations between children some fifty years ago, the Hungarian sociologist S. Szalai observed that any group of about twenty children he checked contained a set of four children any two of whom were friends, or a set of four no two of whom were friends. Despite the temptation to try and draw some behavioral consequences, Szalai realized this may well be a mathematical phenomenon, rather than a sociological one. Indeed, a brief discussion with the mathematicians P. Erdös, P. Turán and V. Sós convinced him this was the case.

Szalai at first thought his finding was sociologically related, but after consulting with mathematicians, discovered that it was actually due to the mathematical properties of networks, rather than how people interact. And with the explosion of social network data now available, this kind of thinking could be done at a new scale. Much work has been done to explore the many properties of large-scale social networks, from the distribution of connections to the average distance from one individual in the network to another. So these researchers (who are also colleagues of mine in the network science community), used a different approach. They examined the nature of smaller graphs in the entire network, and compared this variety to the total possible types of graphs, and in so doing, set out to find “what is a property of graphs and what is a property of people.”

So how does this work? Well, they used a huge amount of Facebook data and constructed three different sets of smaller and denser graphs within the entire network: the first are generated based on the connections between people in a Facebook group, the second based on connections between people attending an event from Facebook, and the third set of graphs is made up of networks derived the connections around individuals. This last type of network is known in social network analysis as an egocentric network, as it’s based on the connection around a single person. For example, if you have ten friends and half of them are connected to each other, this tiny graph will be extracted from the complete network.

By doing this on the complete Facebook network, one gets a very large number of these three different types of miniature networks. Then, they looked to see what structures are in these different networks. Specifically, they looked at the different types of triplets and quadruplets of nodes, or subgraphs, in these smaller networks. For example, when it comes to triplets of nodes, there are four possible ways to connect them: you can have three nodes that are completely connected to each other (a little triangle), totally unconnected, two nodes connected by one edge, or all three nodes connected by only two edges. Since there are only four possibilities, and the fraction of any one type of subgraph in a network is simply one minus the fraction of the other three subgraphs, you can choose three of these triplet subgraphs and graph the relative frequency of them for each little network, as done below:

And this is what they found:

…two striking phenomena already stand out: first, the particular concentrated structure within the simplex that the points follow; and second, the fact that we can already discern a non-uniform distribution of the three contexts (neighborhoods, groups and events) within the space — that is, the different contexts can already be seen to have different structural loci. Notice also that as the sizes of the graphs increases – from 50 to 100 to 200 – the distribution appears to sharpen around the one-dimensional backbone.

But perhaps this non-uniform distribution is simply due to the mathematical constrains of the network, and not due to anything particular to how people interact? Well, through a variety of mathematical models they were able to figure out the rough outer boundaries of this social space—akin to the morphological space above—and then see where each network appears.

Below, they examined the fraction of each subgraph type (for both triads and tetrads, relative to the density of edges in each network). This was overlaid on top of the outer boundaries of the potential social space, which are the light green regions:

As can be seen, the networks only describe a small subset of the total space described by the outer bounds, and the different types of networks describe different regions, meaning that different types of social interactions have different structural, or morphological, properties.

This finding is echoed in a similar result from a paper from about ten years ago, which used full networks and searched for such triads and tetrads within them. Looking for these network motifs, they were able to determine certain hallmarks of distinct classes of networks.

Gratifyingly then, human interactions are far from random and only define a small fraction of the possible space of networks (of which many would be rather implausible social networks), at least when it comes to subgraphs.

But to truly connect morphology to network science, I recommend a research project that examines the social space of mollusc interactions.